Bounds for fidelity of semiclassical Lagrangian states in Kähler quantization

Abstract

We define mixed states associated with submanifolds with probability densities in quantizable closed Kähler manifolds. Then, we address the problem of comparing two such states via their fidelity. First, we estimate the sub-fidelity and super-fidelity of two such states, giving lower and upper bounds for their fidelity, when the underlying submanifolds are two Lagrangian submanifolds intersecting transversally at a finite number of points, in the semiclassical limit. Second, we investigate a family of examples on the sphere, for which we manage to obtain a better upper bound for the fidelity. We conclude by stating a conjecture regarding the fidelity in the general case.